Let k be a positive real number and let A= [ ccc 2 k-1 & 2 k & 2 … - Vidyadip

MCQ

Let $k$ be a positive real number and let $\mathrm{A}=\left[\begin{array}{ccc}2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}0 & 2 k-1 & \sqrt{k} \\ 1-2 k & 0 & 2 \sqrt{k} \\ -\sqrt{k} & -2 \sqrt{k} & 0\end{array}\right]$ If det $(\operatorname{adj} A)+\operatorname{det}(\operatorname{adj} B)=10^6$, then $[\mathrm{k}]$ is equal to $[$ Note : adj $\mathrm{M}$ denotes the adjoint of a square matrix $\mathrm{M}$ and $[\mathrm{k}]$ denotes the largest integer less than or equal to $\mathrm{k}$ ].

A
$4$
B
$6$
✓
$5$
D
$3$

✓

Answer

Correct option: C.

$5$

c
$ |\mathrm{A}|=(2 \mathrm{k}+1)^3,|\mathrm{~B}|=0 \quad(\text { Since B is a skew-symmetric matrix of order 3) } $

$ \Rightarrow \operatorname{det}(\operatorname{adj} \mathrm{A})=|\mathrm{A}|^{-1}=\left((2 \mathrm{k}+1)^3\right)^2=106 \Rightarrow 2 \mathrm{k}+1=10 \Rightarrow 2 \mathrm{k}=9 $

$ {[\mathrm{k}]=4 .}$

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